A Treatise on the Theory of Screws

CHAPTER IX. HARMONIC SCREWS. 104. Definition of an Harmonic Screw. We have seen in § 97 that to each screw 3 of a screw system of the nth order corresponds a screw X, belonging to the same screw system. The relation between 3 and X is determined when the rigid body, and also the screw system which defines its freedom, are completely known. The physical connection between the two screws 3 and X may be thus stated. If an impulsive wrench act on the screw X, the body, if previously quiescent, will commence to move by twisting about 3. We have also seen (§ 102) that to each screw 3 of a screw system of the nth order corresponds a certain screw y belonging to the same screw system. The relation between 3 and y is determined when the rigid body, the forces, and the screw system which defines the freedom, is known. The physical connexion between the two screws 3 and may be thus stated. If the body be displaced from a position of equilibrium by a twist about 3, the evoked wrench, when reduced (§ 96) to the screw system, acts on y. The rigid body being given in a position of equilibrium, and the forces which act on the body being known, and also the screw system by which the freedom of the body is prescribed, we then have corresponding to each screw 3 of the given screw system, two other screws X and y, which also belong to the same screw system. Considering the very different physical character of the two systems of correspondence, it will of course usually happen that the two screws X and r) are not identical. But a little reflection will enable us to foresee what we shall afterwards prove, viz., that when 3 has been appropriately chosen, then X and t/ may coincide. For since n— 1 arbitrary quantities are disposable in the selection of a screw from a screw system of the »th order (§ 69), it follows that for any two screws (for example X and tf) to coincide, n — 1 conditions